PSI - Issue 44

Devis Sonda et al. / Procedia Structural Integrity 44 (2023) 115–122 Devis Sonda et al. / Structural Integrity Procedia 00 (2022) 000–000

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4

a representation of the total absorbed energy of the structure as an area in a force-displacement plane, as illustrated in Fig.1.

Fig. 1. Typical structural frame of industrial buildings and typical force (F) - displacement (S) relationship

The force-displacement relationship is composed of a reversible initial elastic branch and a partly irreversible horizontal plastic branch. Each structure has its own specific graph in which, assuming a global behaviour of an ideal elastic-plastic type, the extension of the ordinates and abscissas can vary but the shape is always the same. By introducing the further simplification of neglecting the deformation associated with elastic behavior, then assuming a rigid-plastic behavior of the system, the area representing energy, is reduced to a rectangle in the force-displacement plane. The equation (9), that expresses the ratio between the total energy of a new equivalent and the existing structure can be represented in a force-displacement plane. The area that represents the ratio between the energy of the real and the reference structure, assumes the shape of a square. In a dimensionless plane (Fig.2), we see that if the existing structure has the same characteristics of the new one then the maximum ordinate and the maximum abscissa assume the unitary value.

P4

F E /F N

1,0

P1

S E /S N

1,0

P3

P2

Fig. 2. Dimensionless force-displacement relationship (E = existing; N = new building) and 45°inclined axes (light blue) In the dimensionless force-displacement plane, we can introduce a new reference system with axes inclined at 45° and see that is possible to represent the same figure through four parameters (P1, P2, P3, P4) different from force and displacement, but which are a combination the two. The area of the square is the dimensionless energy ratio of the two reference structures, the existing and new equivalent one. By assigning the value of a synthesis parameter of seismic capacity to each semi-axis, in terms of force and displacement, we can consider the relationship between the existing structure and the new equivalent structure, obtaining a ratio or, alternatively, a percentage.